A DISTRIBUTED QUANTUM EVOLUTIONARY ALGORITHM WITH A NEW CYCLING OPERATOR AND ITS APPLICATION IN FRACTAL IMAGE COMPRESSION

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1 Inernaional Journal of Arificial Inelligence & Applicaions (IJAIA), Vol.3, No., January 0 A DISTRIBUTED QUANTUM EVOLUTIONARY ALGORITHM WITH A NEW CYCLING OPERATOR AND ITS APPLICATION IN FRACTAL IMAGE COMPRESSION Ali Nodehi, Hosein Mohamadi and Mohamad Tayarani 3 Deparmen of Compuer Engineering, Gorgan Branch, Islamic Azad Universiy alinodehi@gorganiau.ac.ir Deparmen of Compuer Engineering, Azadshahr Branch, Islamic Azad Universiy h_mohamadi@homail.com 3 Deparmen of Compuer Engineering, Souhampon Universiy, Unied Kingdom ayarani@ieee.org ABSTRACT QUANTUM EVOLUTIONARY ALGORITHM (QEA) IS A NOVEL OPTIMIZATION ALGORITHM, PROPOSED FOR COMBINATORIAL PROBLEMS LIKE KNAPSACK AND TRAP PROBLEMS. WHILE FRACTAL IMAGE COMPRESSION IS IN THE CLASS OF NP-HARD PROBLEMS AND QEA IS HIGHLY SUITABLE FOR THE CLASS OF COMBINATORIAL PROBLEMS, QEA IS NOT WIDELY USED IN FRACTAL IMAGE COMPRESSION YET. IN ORDER TO IMPROVE THE PERFORMANCE OF FRACTAL IMAGE COMPRESSION ALGORITHMS, THIS PAPER PROPOSES A DISTRIBUTED QEA WITH A NOVEL OPERATOR CALLED CYCLING QUANTUM EVOLUTIONARY ALGORITHM. IN STANDARD QEA THE DIVERSITY IN THE POPULATION DECREASES ACROSS THE GENERATIONS. DECREASING THE DIVERSITY OF THE POPULATION DECREASES THE EXPLORATION PERFORMANCE OF THE ALGORITHM AND CAUSES THE ALGORITHM TRAPPING IN THE LOCAL OPTIMA. IN THE PROPOSED ALGORITHM, THERE ARE SOME SUBPOPULATIONS SEARCHING THE SEARCH SPACE. AFTER THE SUBPOPULATIONS ARE TRAPPED IN A LOCAL OPTIMUM, THE BEST OBSERVED POSSIBLE SOLUTIONS IN THE SUBPOPULATIONS ARE EXCHANGED IN A CYCLIC MANNER. THE PROPOSED ALGORITHM IS USED IN FRACTAL IMAGE COMPRESSION AND EXPERIMENTAL RESULTS ON SEVERAL IMAGES SHOW BETTER PERFORMANCE FOR THE PROPOSED ALGORITHM THAN GENETIC ALGORITHMS AND QEA. IN COMPARISON WITH CONVENTIONAL FRACTAL IMAGE COMPRESSION, THE PROPOSED ALGORITHM FINDS A SUITABLE SOLUTION WITH MUCH LESS COMPUTATIONAL COMPLEXITY KEYWORDS QUANTUM EVOLUTIONARY ALGORITHM, OPTIMIZATION ALGORITHM, FRACTAL IMAGE COMPRESSION, GENETIC ALGORITHM. INTRODUCTION Fracals exis in naure widely. Fracal images conain few amoun of informaion, bu posses high-level of visual complexiy []. Fracal image compression is poenially a grea coding scheme since i feaures a high compression raio and good rerieved image qualiy. In recen years, many researchers have sudied and improved he fracal image encoding and have goen a lo of achievemens. In 988, he fracal image compression was firsly proposed and uilized a number of affine mappings o denoe he original image. Those mappings are ieraed convergen and heir limi is very close o he original image. In 99, Jacquin inroduced an auomaic DOI : 0.5/ijaia

2 Inernaional Journal of Arificial Inelligence & Applicaions (IJAIA), Vol.3, No., January 0 encoding algorihm for he firs ime ha called baseline fracal image compression or BFC [ 6]. This mehod breaks he original image ino sub-blocks and needs o find he bes mached subblocks according o self-similariies in he image. BFC makes he fracal image encoding become a very hopeful echnique o improve he sorage schemes applied in he consumer elecronics. Alhough BFC is very charming, a grea deal of ime cos during encoding limis i o widely pracical applicaions. In order o solve his problem some improved approaches have been presened. In [6-7] auhors proposed classificaion mehods base on he feaure of domain blocks. [8] proposed a kind of neighborhood maching mehod based on spaial correlaion which makes use of he informaion of mached range blocks and effecively reduced he encoding ime. Fracal Image Compression has araced much ineres due o is high compression raio and qualiy of compressed image. Fracal Image Compression problem is in he class of NP-Hard problems. Quanum Evoluionary Algorihm is a novel opimizaion algorihm proposed for class of combinaorial problems [9]. The probabilisic represenaion of possible soluions in QEA helps he algorihm escaping from local opima. While QEA is highly suiable for class of NP- Hard problems, i is no widely used in Fracal Image Compression. Several works have focused on improving he performance of QEA in solving knapsack and some oher problems. In order o se he parameers of QEA for pracical applicaions, [0] proposes some guidelines for parameer seings. In anoher work, by leading immune conceps in Arificial Immune Sysems, [] proposes a novel algorihm ha is called he immune quanum-inspired evoluionary algorihm. In a more recen work, [] proposes a novel muli-universe parallel Immune QEA ha uses a learning mechanism. The proposed mehod mainains he populaion diversiy along wih beer convergence speed. A muli-agen QEA is inroduced in [3] and he muli-agen operaors, compeiion and cooperaion are performed o. The roaion gae is also considered as leverage o improve he performance of QEA. In [4] a new mehod of calculaing roaion angle of quanum gae is proposed ha causes a rapid convergence for QEA. In order o preserve he diversiy in populaion and empower he search abiliy of QEA, [5] proposes a novel diversiy preservaion operaor for QEA. Reference [6] proposes a sinusoid sized populaion QEA ha makes a radeoff beween exploraion and exploiaion. Several works ry o improve he algorihm of fracal image compression using Geneic algorihm. The low speed of fracal image compression blocks is way o pracical applicaion. In [7] a geneic algorihm approach is used o improve he speed of searching in fracal image compression. A new mehod for geneic fracal image compression based on an eliis model proposed in [8]. In he proposed approach he search space for finding he bes self similariy is grealy decreased. Reference [9] makes an improvemen on he fracal image coding algorihm by applying geneic algorihm. Many researches increase he speed of fracal image compression bu he qualiy of he image will decrease. In [0] he speed of fracal image compression is improved wihou significan loss of image qualiy. In order o preven he premaure convergence of GA in fracal image compression a new approach is proposed in [], which conrols he parameers of GA adapively. A schema geneic algorihm for fracal image compression is proposed in [] o find he bes self similariy in fracal image compression. In evoluionary algorihms he diversiy in he populaion is decreased as generaions advance. When he algorihm reaches is maure sae, he diversiy in he populaion is decreased o is lowes quaniy. In order o deal wih his problem, his paper proposes a disribued QEA wih a 64

3 Inernaional Journal of Arificial Inelligence & Applicaions (IJAIA), Vol.3, No., January 0 cycling exchange of bes observed possible soluions. The proposed algorihm is applied on fracal image compression problem and shows beer performance for he proposed algorihm han convenional QEA and GA. This paper is organized as follows, in Secion, he heoreical basics of fracal image compression is presened. In secion 3, QEA and is represenaion is presened, Secion 4 proposes he novel disribued QEA wih he cycling operaor, in Secion 5 he proposed algorihm is applied on fracal image compression. Several experimenal resuls are performed o in Secion 6 o es he proposed algorihm, and finally Secion 7 concludes he paper.. THEORETICAL BASICS OF FRACTAL IMAGE COMPRESSION The fracal image compression is based on he local self-similariy propery and PIFS. The relaed definiions and heorems are saed as follows [5-6]: Definiion.. Le X be a meric space wih meric d x. A map w:x X is Lipschiz wih Lipschiz facor s, if here exiss a real value s such ha dx( w( x), w( y)) sdx( x, y), 0 s, x, y X () We also say ha w is conracive wih conraciviy s. Definiion.. Le X be a meric space and d x be is meric. For a poin x X and a nonempy se A X, le us firs define he disance of x o A by d ( x, A) = infd ( x, a) () y a A Then he Hausdorff disance beween A and B is defined for any nonempy ses A, Where x B X as d( A, B ) = Max( d ( A, B ), d ( B, A )) (3) d h h h ( A, B ) = sup d ( a, B ) (4) a A Le X be he se of N N gray level images. The meric is defined as he usual Euclidean disance by regarding he elemens in X as vecors of dimension N N. Le I be a given image belonging o X. The goal is o find a se of ransformaions { w, w,..., wn }, each of which is a resriced funcion and saisfies (), such ha he given image I is an approximae aracor. The se w, w,..., w } is called PIFS. The following heorem is an imporan fac for PIFS. { n Theorem.. Consider a PIFS w, w,..., wn wih w i : X X for all i. Le W =Uw i. Then here exiss a unique poin A X such ha for any poin B X Y n A = W( A ) = Lim w ( B ) (5) n The poin A in (5) is called he fixed poin or he aracor of he mapping W. Nex, he famous Collage heorem will be inroduced. Theorem.. Le { w, w,..., wn } be a PIFS wih conraciviy facor s. Le B be any nonempy compac se in X. Then we have d( A, B ) ( s ) d( B, W( B )) (6) where A is he aracor of he mapping W and d is he Hausdorff meric (3). 65

4 Inernaional Journal of Arificial Inelligence & Applicaions (IJAIA), Vol.3, No., January 0 Le d ( I, W( I )) ε where e is a very small posiive real number. By he Collage heorem, one can obain ha ε d ( A, I ) (7) ε From Eq. (7), one can see ha afer a large number of ieraions, an aracor A is generaed which is sufficienly close o he given image I. For pracical implemenaion, le I be a given gray level image. The domain pool D is defined as he se of all possible blocks of size 6 6 of he image f, which makes up (56-6+) (56-6+) =5808 blocks. The range pool R is defined o be he se of all nonoverlapping blocks of size 8 8, which makes up (56/8) (56/8) = 04 blocks. For each block v from he range pool, he fracal ransformaion is consruced by searching all elemens in he domain pool D he mos similar block. Le u denoe a sub-sampled domain block which is of he same size as v. The similariy of u and v is measured using Mean Square Error (MSE) defined by MSE= J 0 I 0 [ u( i, j ) v( i, j )] The fracal ransformaion allows he dihedral ransformaion of he domain blocks, i.e., he eigh orienaions of he blocks generaed by roaing he blocks counerclockwise a angles 0, 90, 80, and 70 degrees and flipping wih respec o he line y = x, respecively. Thus, for a given block from he range pool, here are = 464,648 MSE compuaions o obain he mos similar block from he domain pool. Thus, in oal, one needs ,648 = 475,799,55 MSE compuaions o encode he whole image using his full search compression mehod. For a given range block v, he fracal ransformaion also allows he adjusmen of he conras p and he brighness q on he subsample domain block u. The similariy is measured by he quaniy d = p. uk + q v, where u k, 0 k 7,are he eigh orienaions of u. By simple opimizaion mehod, p and q can be compued direcly as (8) And [ N u, v u, v, ] p = (9) [ N u, u u, ] q = N.[ v. p u. ] (0) Where N=64. The posiion of he mos similar domain block, he conras p, he brighness q, and he orienaion k consiue he fracal code of he given range block v. In pracice, for image, 6 bis are required o represen he posiion of he domain block. Finally, as v runs over all 04 range blocks in he range pool R, he encoding process is compleed. To decode, one firs makes up he 04 affine ransformaions from he compression codes and chooses any iniial image. Nex, one performs he 04 affine ransformaions on he image o obain a new image, and hen proceeds recursively. According o Theorems. and., he sequence of images will converge. The sopping crierion of he recursion is designed according 66

5 Inernaional Journal of Arificial Inelligence & Applicaions (IJAIA), Vol.3, No., January 0 o user s applicaion and he final image is he rerieved image of fracal coding. 3. QUANTUM EVOLUTIONARY ALGORITHMS QEA is inspired from he principles of quanum compuaion, and is superposiion of saes is based on q-bis, he smalles uni of informaion sored in a wo-sae quanum compuer [8,0]. A q-bi could be eiher in sae 0 or, or in any superposiion of he wo as described below: ψ = α 0 + β () Where α and β are complex number, which denoe he corresponding sae appearance probabiliy, following below consrain: α + β = () This probabilisic represenaion implies ha if here is a sysem of m q-bis, he sysem can m represen saes simulaneously. A each observaion, a q-bis quanum sae collapses o a single sae as deermined by is corresponding probabiliies. 3. Represenaion QEA uses a novel represenaion based on he above concep of q-bis. Consider ih individual in h generaion defined as an m-q-bi as below: q i α i αi αii αim = ( 3) βi βi βii βim Where α + β =, j=,,,m, m is he number of q-bis, i.e., he sring lengh of he q-bi ij ij individual, i=,,,n, n is he number of possible soluion in populaion and is generaion number of he evoluion. Since a q-bi is a probabilisic represenaion, any superposiion of saes is simulaneously represened. If here is, for insance, a hree-q-bis(m = 3) individual such as (4): 3 q = i (4) 3 3 Or alernaively, he possible saes of he individual can be represened as: i q = (5) 6 3 Noe ha he square of above numbers are rue probabiliies, i.e. he above resul means ha he probabiliies o represen he sae 000, 00, 00, 00 are /4, /8, /4 and / respecively. Consequenly, he hree-q-bis sysem of (4) has all eigh saes informaion a he same ime. Evoluionary compuing wih he q-bi represenaion has a beer characerisic of diversiy han classical approaches since i can represen superposiion of saes. Only one q-bi individual such as (4) is enough o represen eigh saes, whereas in classical represenaion eigh individuals are needed. Addiionally, along wih he convergence of he quanum individuals, he diversiy will gradually fade away and he algorihm converges

6 Inernaional Journal of Arificial Inelligence & Applicaions (IJAIA), Vol.3, No., January 0 3. QEA Srucure T In he iniializaion sep of QEA, [ αij βij ] of all q0i are iniialized wih. This implies ha 0 each q-bi individual q i represens he linear superposiion of all possible saes wih equal probabiliy. The nex sep makes a se of binary insans; x i by observing Q ( ) = { q, q,..., q n } saes, where X ( ) = { x, x,..., xi,..., xn } a generaion is a random insan of q-bi populaion. Each binary insan, x i of lengh m, is formed by selecing each bi using he probabiliy of q-bi, eiher α ij or β ij of q i. Each insan x i is evaluaed o give some n measure of is finess. The iniial bes soluion b = max{ f ( xi )} is hen seleced and sored from i= among he binary insans of X(). Then, in updae Q(), quanum gaes U updae his se of q-bi individuals Q() as discussed below. This process is repeaed in a while loop unil convergence is achieved. The appropriae quanum gae is usually designed in accordance wih problems under consideraion. The pseudo-code of QEA algorihm is defined as [4]: Procedure QEA begin =0. iniialize Q(0).. make X(0) by observing he saes of Q(0). 3. evaluae X(0). 4. Sore X(0) ino B(0). Sore he bes soluion among X(0) ino b. 5. while no erminaion condiion do begin =+ 6. make X() by observing he saes of Q(-) 7. evaluae X() 8. updae Q() using Q-gaes 9. sore he bes soluions among B(-) and X() ino B() 0. sore he bes soluion among B() ino b. if global migraion condiion hen migrae b o B() globally. else if local migraion condiion hen migrae b k in B() o B() locally end end QEA has a populaion of quanum individuals ( ) q, q,..., q } Q = { n, where is generaion sep and n is he size of populaion. The QEA procedure is described as: 0 αij. In he iniializaion sep all q-bis 0 and ij i =,,..., n and j =,,..., m are iniialized wih.. I means he probabiliy of observing "0" and "" for all q-bis is equal.. In his sep he binary soluions observing Q(0). Observing xij from qubi x ij X (0) = { x 0 0 0, x,..., xn T ij ij ] [ α β 0 if R(0,) < α ij = oherwise } a generaion = 0 are creaing by is performed as below: Where R (, ), is a uniform random number generaor. 3. All soluions in X() are evaluaed wih finess funcion. 4. Sore X(0) ino B(0). Selec bes soluion among X(0) and sore i o b. (6) 68

7 Inernaional Journal of Arificial Inelligence & Applicaions (IJAIA), Vol.3, No., January 0 5. The while loop is running unil erminaion condiion is saisfied. Terminaion condiion can be considered as maximum generaion condiion or convergence condiion. 6. Observing X() from Q(-). 7. Evaluae X() by finess funcion 8. Updae Q() 9, 0. Sore he bes soluions among B(-) and X() o B(). If he fies soluion among B() is fier han b hen sore he bes soluion ino b.,. If global migraion condiion is saisfied copy b o all he soluions in B(). If local migraion condiion is saisfied replace some of soluions in B() wih bes one of hem. 3.3 Quanum Gaes Assignmen The common muaion is a random disurbance of each individual, promoing exploraion while also slowing convergence. Here, he quanum bi represenaion can be simply inerpreed as a biased muaion operaor. Therefore, he curren bes individual can be used o seer he direcion of his muaion operaor, which will speed up he convergence. The evoluionary process of quanum individual is compleed hrough he sep of updae Q(). A crossover operaor, quanum roaion gae, is described below. Specifically, a q-bi individual q i is updaed by using he roaion gae U(θ) in his algorihm. The jh q-bi value of ih quanum individual in T generaion [ α β is updaed as: ij ij ] + α ij cos( θ ) sin( θ ) α ij = + (7) β ij sin( θ ) cos( θ ) β ij Where θ is roaion angle and conrols he speed of convergence and deermined from Table I. Reference [8] shows ha hese values for θ have beer performance. TABLE I. Lookup able OF θ. xi b i f ( x) f ( b) θ 0 0 false rue 0 0 false 0.0π 0 rue 0 0 false 0.0π 0 rue 0 false 0 rue 0 4. DISTRIBUTED QEA WITH CYCLING OPERATOR In QEA when he algorihm converges, all he q-individuals reach similar values and heir quanum bis converge o eiher or 0. This means ha here is lile if any exploraion during laer sages of he algorihm s convergence and ha all of he individuals search abou a single opimum before converging o i. In oher words, he algorihm canno explore he oher pars of 69

8 Inernaional Journal of Arificial Inelligence & Applicaions (IJAIA), Vol.3, No., January 0 he search space well, and all of he individuals are exploiing a single opimum. This secion proposes a novel Disribued QEA wih a cycling operaor o improve he diversiy of he populaion and so he performance of he QEA. In he proposed algorihm he populaion is divided ino some dependen subpopulaions searching he search space. Since each subpopulaion searches he search space independenly, each of hem reaches a differen local opimum. Here convergence esimaion is used o deermine if he subpopulaion is rapped in a local opimum. The converged subpopulaions are considered as he rapped subpopulaions and he cycling operaor will be performed on hem. In he cycling operaor, he bes observed of he subpopulaions will be exchanged. When he bes observed possible soluions are exchanged, he q-individuals of each subpopulaion will move oward anoher local opimum. During his movemen, he q-individuals have a chance o mee new local opima and i increases he chance of possible soluions in finding global opimum. The algorihm of he proposed algorihm is briefly shown in figure. The propose algorihm is similar o excep in seps 4 and 6. In his sep he diversiy preserving operaor is performed on all he q-individuals. In sep 4 he bes possible soluion in each subpopulaion is found and he bes possible soluion hroughou he search is sored in Bl. All he q-individuals in each subpopulaion search around Bl. The disribued naure of he proposed algorihm prevens he whole populaion from rapping in local opima. Alhough here is a chance for each of he subpopulaion rapping in local opima, he saisical naure of QEA guaranees he subpopulaions searching disinc local opima. Afer rapping in local opima, he cycling operaor is applied on he subpopulaions. Here convergence esimaion is used o deermine if he subpopulaion is rapped in a local opimum. The q-individual q i converges if i saisfies he below consrain [3]: m i, k m k = α > γ Where, m is he size of q-individuals. This consrain is saisfied only when he average disance beween he q-bis of he q-individual or [ β ] T = [ 0 ] T q i (8) and he converged saes ( [ α β ] T = [ 0 ] T α ) is less han γ. In sep 6, when he subpopulaions are converged, he bes observed possible soluions are exchanged among he subpopulaion in a cyclic manner: - B S = B - For l= o S- 3- B i = B i+ Where B is he bes observed possible soluion of he h subpopulaion and S is he number of subpopulaions. 70

9 Inernaional Journal of Arificial Inelligence & Applicaions (IJAIA), Vol.3, No., January 0 5. DISTRIBUTED QEA FOR FRACTAL IMAGE COMPRESSION The proposed algorihm for each range block, searches among all he domain pool o find he bes domain block and he bes ransformaion. The coding mehod for he q-individuals in he proposed mehod is as figure. P x P y P Figure. The coding of he addresses for domain blocks. P x is he horizonal posiion of domain block, P y is he verical posiion of domain block and P is he ransformaion. In he proposed approach each q-individual, has hree pars: P x shows he horizonal posiion of domain block, P y shows he verical posiion of he domain block and P shows he ransformaion. The ransformaions are he 8 ordinary ransformaions: roae 0, 90, 80, 70, flip verically, horizonally, flip relaive o 45, and relaive o 35. Each par of each soluion is a real number and is convered o an ineger number before evaluaion process. Figure. The algorihm of he propose algorihm. 7

10 Inernaional Journal of Arificial Inelligence & Applicaions (IJAIA), Vol.3, No., January 0 6. EXPERIMENTAL RESULTS This secion experimens he proposed algorihm and compares he proposed algorihm wih he performance of GA in fracal image compression. The proposed algorihm is examined on images Lena, Pepper and Baboon wih he size of and gray scale. The size of range blocks is considered as 8 8 and he size of domain blocks is considered as 6 6. In order o compare he qualiy of resuls, he PSNR es is performed: Where m n is he size of image. PSNR= 0 log n m m n i= j= 55 ( f ( i, j) g( i, j) ) The crossover rae in GA is 0.8 and he probabiliy of muaion is for each allele. he parameers of QEA is se o Table I. The size of subpopulaions is se o 5 for all he experimens. The convergence crieria, γ is considered as Table II shows he experimenal resuls on he proposed algorihm and GA. The number of ieraions for GA, QEA and he proposed algorihm for all he experimens is 00. According o able II he proposed algorihm improves he performance of fracal image compression for all he experimenal resuls. (9) 7

11 Inernaional Journal of Arificial Inelligence & Applicaions (IJAIA), Vol.3, No., January 0 TABLE II. Experimenal resuls on Lena, Pepper and Baboon Picure Lena Pepper Baboon Mehod Populaion MSE Size compuaions PSNR Full Search - 59,474, ,44, QEA 5 5,0, ,096, ,07, ,44, PROPOSED 5 5,0, ALGORITHM 0 4,096, ,07, ,44, GA 5 5,0, ,096, ,07, Full Search - 59,474, ,44, QEA 5 5,0, ,096, ,07, PROPOSED ALGORITHM GA 30 6,44, ,0, ,096, ,07, ,44, ,0, ,096, ,07, Full Search - 59,474, ,44, QEA 5 5,0, ,096, ,07, PROPOSED ALGORITHM GA 30 6,44, ,0, ,096, ,07, ,44, ,0, ,096, ,07,

12 Inernaional Journal of Arificial Inelligence & Applicaions (IJAIA), Vol.3, No., January 0 (a) (b) (c) (d) (e) (f) Figure 3. (a) Original image Lena of size 56 56,(b) iniial image for decoder, (c) full search mehod, (d) proposed mehod, (e) GA mehod, (e) QEA mehod 7. CONCLUSION Quanum Evoluionary Algorihms are novel algorihm proposed for combinaorial opimizaion problems like knapsack problem. Since fracal image compression is in he class of NP-Hard problems, QEA is highly suiable for his problem bu is no properly applied o fracal image compression ye. This paper proposes a Disribued QEA wih a cycling Operaor for QEA in fracal image compression. In evoluionary algorihms during he search process he diversiy in he populaion is decreased, and he algorihm is rapped in he local opima. This is rue for QEA which uses a probabilisic represenaion for he individuals. Here we propose a new mehod o help he algorihm escaping from he local opima. In he proposed algorihm he populaion is divided ino some subpopulaions and a cycling operaor exchanges he bes observed possible soluions among he subpopulaions. The experimenal resuls on Lena, Pepper, and Baboon picure show an improvemen on fracal image compression. 74

13 Inernaional Journal of Arificial Inelligence & Applicaions (IJAIA), Vol.3, No., January 0 ACKNOWLEDGEMENTS The auhors would like o reurn hanks he financial suppor he Islamic Azad Universiy, Gorgan Branch, Iran. REFERENCES [] Peigen H.O., Jurgens H. and Saupe D., Chao and Fracals: New Froniers of Science, Springer Verlag, New York, 99. [] Barnsley, M.F., Fracal Everywhere, Academic Press, California, 988. [3] Barnsley, M.F. and Demko, S., Ieraed funcion sysems and he global consrucion of fracals. Proceedings of he Royal Sociey, PP 43-75,985. [4] Barnsley M.F., Elon J.H. and Hardin D.P, Recurren ieraed funcion sysems Consr., Approx. 3 3, 989. [5] Jacquin A.E, coding based on a fracal heory of ieraed conracive image ransformaions, IEEE Trans. Image Process,PP 8-30, 99. [6] Fisher Y, Fracal Image Compression: Theory and Applicaion, Springer Verlag, Berlin, 995. [7] Wang Z., Zhang D. and Yu Y., Hybrid image coding based on parial fracal mapping, Signal Process. Image Commun, pp , 000. [8] Truong T.K, Kung C.M and Jeng J.H., Hsieh M.L., Fas fracal image compression using spaial correlaion, Chaos Soliions Frac, Chaos Soliions Frac, pp , 004. [9] K. Han, J. Kim, Quanum-inspired evoluionary algorihm for a class of combinaorial opimizaion, IEEE Trans. on Evoluionary Compu, vol., no. 6, 00. [0] Kuk-Hyun Han Jong-Hwan Kim "On seing he parameers of quanum-inspired evoluionary algorihm for pracical applicaion", IEEE Congress on Evoluionary Compuaion, 003. [] Ying Li Yanning Zhang Rongchun Zhao Licheng Jiao, "The immune quanum-inspired evoluionary algorihm ", Inernaional Conference on Sysems, Man and Cyberneics, 004 IEEE. [] Xiaoming You, Sheng Liu and Dianxun Shuai "On Parallel Immune Quanum Evoluionary Algorihm Based on Learning Mechanism and Is Convergence", Springer, Advanced in Naural Compuaion [3] Chaoyong Qin Jianguo Zheng and Jiyu Lai, "A Muliagen Quanum Evoluionary Algorihm for Global Numerical Opimizaion", Springer, Life Sysem Modeling and simulaion, Vol 4689, pp , 007. [4] Rui Zhang Hui Gao, "Improved Quanum Evoluionary Algorihm for Combinaorial Opimizaion Problem", IEEE Inernaional Conference on Machine Learning and Cyberneics, 007. [5] M.-H. Tayarani- N. and M.- R. Akbarzadeh-T A Cellular Srucure and Diversiy Preserving operaor in Quanum Evoluionary Algorihms. IEEE World Conference on Compuaional Inelligence 008. [6] M.-H. Tayarani- N. and M.- R. Akbarzadeh-T A Sinusoid Size Ring Srucure Quanum Evoluionary Algorihm. IEEE Inernaional Conference on Cyberneics and Inelligen Sysems Roboics, Auomaion and Mechanics 008. [7] Xiao Chen, Guang-Xi Zhu, and Yaoing Zhu, Fracal image coding mehod based on geneic algorihms Inernaional Symposium on Mulispecral Image Processing 998. [8] Mira, S.K. Murhy, C.A. Kundu, M.K Technique for fracal image compression using geneic algorihm, IEEE Trans on Image Processing Vol 7 no 4 pp [9] Lu Xun Yu Zhongqiu The applicaion of GA in fracal image compression, 3rd IEEE World Congress on Inelligen Conrol and Auomaion, 000. [0] Gafour, A. Faraoun, K. Lehireche, A Geneic fracal image compression. ACS/IEEE Inernaional Conference on Compuer Sysems and Applicaions, 003. [] Lifeng Xi, Liangbin Zhang, A Sudy of Fracal Image Compression Based on an Improved Geneic Algorihm, Inernaional Journal of Nonlinear Science Vol 3, No, pp 6-4m

14 Inernaional Journal of Arificial Inelligence & Applicaions (IJAIA), Vol.3, No., January 0 [] Ming-Sheng Wu, Jyh-Horng Jeng, and Jer-Guang Hsieh, Schema geneic algorihm for fracal image compression Elsevier Journal of Engineering Applicaions of Arificial Inelligence Vol 0, Issue 4, pp [3] K. Han, J. Kim, Quanum-Inspired Evoluionary Algorihms wih a New Terminaion Crierion, H Gae, and Two-Phase Scheme, IEEE Trans. on Evoluionary Compuaion, vol. 8, no.,